Achieving Mediated Agreements using

Agreement Space Modeling

C. Carrascosa, M. Rebollo

Universidad Politecnica de ValenciaDepartamento de Sistemas Informaticos y Computacion (DSIC)

Camino de Vera s/n – 46022 Valencia – Spain{carrasco, mrebollo}@dsic.upv.es

Abstract. An agreement is an arrangement between two or more enti-ties to do something. This implies a context in which terms the agreementis developed. This paper presents a model of such agreement context asan euclidean space. On the other hand, an agreement can also be seenas a Constraint Satisfaction Problem (CSP) which solution space modelsan agreement space. This will allow to have a mediator that could notonly model the agreement as an space, but also to check the viability ofthe agreement as it is being built. This mediator, that could even givesome counsels about the agreement terms being formulated, is called hereCounselor Agent. The protocols to interact with such agent to build anagreement are also presented.

1 Introduction

According to the Merrian-Webster dictionary, agreement is “an arrangement asto a course of action. Compact, treaty a: a contract duly executed and legallybinding b: the language or instrument embodying such a contract”. This agree-ment, this contract, assumes or implies a context in which it is going to becarried out. So, using a classical example by Castelfranchi [3] regarding the wayan speech act tries to induce different motives for goal adoption in the addressee,when a general commands to a soldier ’Fire!’, the soldier could interpret suchan order in different ways according to the several meanings of such expression(he could shoot, or offer to the general some matches, or look for the firemen,or burn a fire, ...), but the most probable outcome is that he wouldn’t doubtbetween such options and shoot his weapon, due to the context of the situation,that is, there is an implicit agreement between general and soldier regardingorders and military concepts.

This paper continues the work of [2][1] that define an agreement by means ofa multi-dimensional Euclidean space approach. This modeling will allow to usegeometrical properties or operations to work with such agreements, and evento model the agreement problem as a constraint-satisfaction problem (CSP)and to solve it using the Hyperpolyhedron Search Algorithm (HSA 6=) [11].This paper focuses in the way to use this modeling as a CSP by a mediator(Counselor Agent), which could be able to on-line evaluate the feasibility of such

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an agreement, or even give some advises as the agreement negotiation evolves.In order to do that, several agreement-related interaction protocols have beendefined to be used by the participant agents to reach an agreement upon a setof agreed terms.

So, one of the main differences between this approach and previous worksrelated with agreements and agents, is the focus of the approach. In works asContract-related agents [8], the focus is in the agent, and the way he may dealwith the concept of contract. The work here presented focuses in the concept ofagreement as an upper abstraction over the idea of agent, organization or otherentity. Modeling the agreement in this way, is what can be really useful to acounselor mediating in such agreement.

The rest of the paper is structured as follows. Context Spaces are introducedin section 2. Section 3 provides a set of general definitions needed to describe theproposed protocols and algorithms. Agreement-related interaction protocols aredescribed in section 4 and section 5 explains how the Counselor Agents worksto help participant agents to reach an agreement. A simple example is shown insection 6 and, finally, section 7 summarizes this approach and concludes.

2 Context Space

This term is used in the pervasive computing field related to context-awarecomputing [10] [9] to define a theory that models contexts based on intuitionsfrom state-space models. In this way, the context space of an application isdetermined by the types of information, deemed relevant and obtainable bythe designers, that is the context attributes, and the domain of possible valuesin each attribute. Each one of these attributes is considered one dimension ina multidimensional Euclidean space. This multi-dimensional space defines thespace in which context can be perceived.

This approach also includes a context algebra to express situations in termsof Context Spaces and to reason about such situations by means of a set ofoperators and calculations. The operators defined are the scalar difference (itcalculates the degree of similarity between two comparable context states), andthe intersection (it produces a new context space containing shared regions ofvalues of the same attributes between two comparable context spaces).

By representing situations and system’s state as multi-dimensional objects,it is possible to generically describe and consequently reason about context ofa system. A region of acceptable values is defined as a set which satisfies somepredicate, hence, it can consist of any information (numerical or non-numerical)that best reflect the context attribute behavior (in terms of possible values) forthe specific situation.

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3 Agreements Concepts

This section presents some generic concepts and definitions related to agreementsneeded to explain the algorithms and protocols designed for a counselor agent toguide the agreement process. A more detailed definition can be found in [2][1].

3.1 Agreement Definitions

An agreement is the definition of a working context for two or more beingsor entities (agents, organizations, . . . ) so that it is obtained as the result of anegotiation process.

Definition 1 (Agreement). An agreement Ag is defined as Ag = (E, Cx),where:

– E = {E1, E2, . . . En} is a set of entities participating in Ag, so that ∀Ei, i =1, . . . n,∃!Oi = {oij}, j = 1, . . .m, ontology or set of concepts known by Ei

– Cx = {(cxo, cxIo)|cxo ∈

⋃i Oi, cx

Io ⊆ Do}, where Do is the domain of the

ontology term cxo. Thus, this context is formed by a set of ontological termscxo with its corresponding set of valid instances cxI

o that have been agreed byE.

Definition 2 (Agreement Discourse Universe). The Agreement DiscourseUniverse of an agreement Ag –ADU(Ag)– is the whole set of concepts knownby all the entities participating in the agreement. So, if Oi is the set of all theontologies known by the entity Ei participating in the agreement Ag (i = 1..n),then D = {o/o ∈ Oi △ Ok, ∀i 6= k/i, k = 1..n} (the symmetric difference of allthe agreement participating entities’ ontologies),1 and the ADU(Ag) is definedas ADU(Ag) =

⋃i Oi − D.

The ADU(Ag) is formed by all the concepts that at least one pair of entitiesparticipating in the agreement Ag share (Figure 1). If this set is empty, entitieshave not anything in common and they can not reach any kind of agreement.

3.2 Agreement Process

An agreement Ag has two phases:

1. Reaching an agreement (defining the agreement context Cx): It comprisesthe negotiation process between two or more entities (belonging to E) toreach an agreement. In fact, it is decided in two levels:

(a) A decision must be taken about what are the concepts around whichsuch agreement is going to be related, that is, Cx ⊆ ADU(Ag).

1 The symmetric difference is defined as A△B = (A∩B′)∪(A′∩B), where A

′ denotesthe complementary set of A

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Fig. 1. Agreement Discourse Universe and Agreement Discourse Space with the pro-jection to a 2-dimensional Agreement Space

(b) After that, the specific terms of such agreement must be fixed, that is,the values or intervals for the concepts in Cx.

2. Agreement execution: In this phase each entity must fulfill the accomplishedagreement executing the needed actions or calculus according to the contextdefined by such agreement. This execution could not even imply any kind ofadditional coordination.

3.3 Agreement Space

As it has been stated before, an agreement can be seen as the definition of acommon context. As commented in section 2 an space metaphor such as the oneused in Context Spaces can be used to express such contexts. So, an agreementspace can be defined.

Definition 3 (Agreement Discourse Space). The Agreement Discourse Spaceof an agreement Ag ADS(Ag) is defined by considering as a dimension (in anEuclidean space) each concept included in an Agreement Discourse Universe ofan agreement Ag. That is, the ADS(Ag) is an n-dimensional space, where n isthe cardinality of the ADU(Ag) (number of common terms). Figure 1 shows anexample with 3 dimensions, where the points represent the possible combinationvalues of such space (assuming a discrete domain for every dimension).

Definition 4 (Agreement Space). According to the first phase of an agree-ment, a decision must be taken about the concepts, that is, the dimensions inwhich the agreement will be expressed. So, an Agreement Space is a projectionof the Agreement Discourse Space onto the dimensions defining the agreement.That is, this space will be defined by the features the different entities Ei makingthe agreement are going to negotiate (Cx), each one of such features defining adimension in this space (∀i : di ∈ dim(Ei, Ag)).

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In order to be possible an agreement Ag, for all entity Ei participating inthe agreement there will exist at least one other participating entity Ej so thatdim(Ei, Ag) ∩ dim(Ej , Ag) 6= 0.

The Agreement Space is, then, the result of eliminating the unnecessarydimensions that are not relevant for the agreement. For example, in Figure 1the y dimension is not relevant, so the way the entities will deal with y conceptis not controlled by the agreement, and the Agreement Space is reduced to themarked 2-dimensional area, that is, the projection over x and z dimensions ofthe defined ADS.

In this way, the outcome of the first phase of the agreement will be thedefinition of this Agreement Space, fixing the satisfying values for each one of thedifferent dimensions comprising this space. The second phase of the agreementwill be to carry out the execution of the agreement taking into account that ithas to be carried out inside the previously defined Agreement Space.

Definition 5 (Local Agreement Space). The Local Agreement Space forentity Ei in the agreement Ag is defined as the projection over the dimensionsof interest of entity Ei in such Agreement Space.

4 Agreement-Related Interaction Protocols

WS-Agreement [6] is a standard proposed by the Global Grid Forum (GGF)that includes the definition of a simple interaction protocol to support one-to-one negotiation, with the likely aim to support different mechanisms in thefuture through definition of multiple interaction protocols. So, the work by [7]for what they called service agreement where one consumer of a service choosesone service provider from n available by means of the FIPA Iterated ContractNet Interaction Protocol[5].

In this proposal, a Counselor Agent (see Section 5) mediates in the agreementprocess in order to help the participant agents to reach an agreement related to acommon vocabulary. The mediation is achieved though a set of services availableto all participants that help to find a common Agreement Space, inside whichagents can interact being ensured that the terms of the agreement are fulfilled.

This section presents two new interaction protocols that has been definedto deal with the services offered by the Counselor Agent: ADU Definition andAgreement Counselor, that will be addressed by the ADU Interaction Protocoland the Mediated-Agreement Interaction Protocol, respectively.

4.1 ADU Interaction Protocol (ADUIP)

This interaction protocol controls the service of the Counselor Agent related tothe definition of the Agreement Discourse Universe of a group of entities, that is,the common vocabulary shared by such entities (see Definition 2). This protocoldescribes a general phase, previous to the agreement itself, that can be used inmany other situations in which establish a common vocabulary is needed. Figure2 shows this protocol, that is composed of the following steps:

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Initiator Counsellor Participant i

call for counsellor

refuse

agree

not-understood

failure-no-match

call for agreement

refuse

agree

not-understood

failure-proxy

inform

m

n

j <= n

k <= n - j

l = n - j - k

request

refuse

agree

not-understood

p

r <= p

s <= p - r

t = p - r - s

p

t

t

t

deadline

Fig. 2. ADU Interaction Protocol

1. An Entity Ei sends a Call For Counselor to all entities offering the serviceof Counselor indicating an AgreementID.

2. If any Counselor agrees to Ei, then Ei will send a Request for Counselor toone of them, Cj , that will serve as counselor for agreement AgreementID.

3. Counselor Cj sends a Call For Agreement to all entities in the system, in-dicating an Agreement ID and a deadline to this phase of joining to theagreement.

4. Each entity Ek that wants to participate in the agreement will send to Cj

an agree message with his ontology before deadline is met.

5. After deadline, Cj will calculate the ADU using the ontologies received fromthe entities that will participate in the agreement, communicating them ifthe ADU is empty that there is a failure, because there is no possibility ofagreement.

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The ADU Interaction Protocol is only focused on provide a framework toparticipant agents for exchanging their ontologies (or fragments of them). Itdoes not try to solve the ontology alignment problem, which is out of the scopeof this paper.

4.2 Mediated-Agreement Interaction Protocol

After the existence of a common vocabulary is ensured by ADU protocol, theparticipant entities can negotiate the terms of any agreement. A two-phasesprotocol is proposed to reach an agreement among a group of entities (agentsor organizations) mediated by a counselor. The final goal of this protocol is toconstruct the Agreement Space by means of the following steps:

1. Defining the context: It will be formed by the dimensions of the AgreementSpace, that is, the concepts that will be dealt in the agreement, and that willbe part of the ADU (in fact, this context will be a projection of the ADU inthe dimensions of the Agreement Space).

2. Defining the agreement terms, that is the different values or intervals for thedimensions defining the context that will define the Agreement Space. Thisprocess will be done by applying one by one the constraints suggested by theparticipants. Moreover, it is regulated by a mediator—the Counselor—whichguides the negotiation process to achieve an accord among the participants.Its main task is to check that any additional constraint is consistent withthe current constraint set. That is, if they formed a convex hull.

The protocol here presented is described from the Counselor point of view,and, its purpose is to help in the Agreement Space definition. This protocol (seeFigure 3a) is divided into two main parts: the first one deals with the definitionof the context, and the second one with the definition of the agreement terms(see Figure 3b).

1. Context Definition: An initiator requests the Counselor for an agreement.

(a) An Entity Ei sends a Call For Agreement with AgreementID to theCounselor Agent that has previously calculated the ADU, indicatingthat he wants to begin an agreement with some or all the other entitiesinvolved in such ADU.

(b) If the Counselor agrees to Ei, then he sends a Call For Context to allthe entities associated to the previous ADU, indicating an Agreement IDand a deadline to this phase of joining to the agreement.

(c) Each entity Ek that wants to participate in the agreement will send tothe Counselor an Agree message with his agreement dimensions (that is,the concepts he is interested in negotiate for the agreement, and that arepart of the previous ADU) before deadline is met.

2. Agreement Terms Definition: The former initiator is turned into anotherparticipant and the Counselor takes the initiative.

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Initiator Counsellor Participant i

call for agreement

refuse

agree

not-understood

call for context

refuse

agree

not-understood

agreement terms

subprotocol

proxied-communicative-act

reply-message

p

r <= p

s <= p - r

t = p - r - s

p

Counsellor Participant i

inform-context p

call-for-terms p

inform-term

[redundant]reject

[¬consistent]reject

[consistent]accept

p

inform-done

failure

call-for-terms

(a) (b)

Fig. 3. (a) Mediated-Agreement Interaction Protocol (b) Agreement terms definitionsubprotocol

(a) The Counselor Informs to all the entities of the dimensions of the agree-ment (from the general ADU, different concrete agreements could takeplace, between different or the same entities).

(b) The Counselor sends a Call For Terms message to all the participatingentities with the proper AgreementID to ask for agreement terms.

(c) Each one of the participating entities will send an Inform message tothe Counselor for each one of the negotiation terms (in the form of aconstraint) that he is interested in. Such messages will be processed bythe Counselor to build the AS.

(d) For each Inform message, the Counselor will respond either with a Rejectmessage, if the new term is not consistent with the agreement or isredundant, or with an Accept message if the new term is consistent.

(e) The process of receiving and analyzing new agreement terms continuetill there is no more terms (according to a deadline-based method).

(f) The Counselor informs the participating entities either the final agree-ment terms, or that the agreement has not been possible.

The final result is the definition of the Agreement Space, modeled as a hyper-polyhedron formed by all accepted constraints (agreement terms). During theexecution of the agreement, all interactions have to be inside this space as theparticipants have agreed.

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5 Using Agreement Spaces for Mediators

It is not so strange, when trying to reach an agreement to use a mediator to,not only watch over the fulfillment of the different processes involved in reachingan agreement, but also to counsel about the final values or intervals of theagreement due to its global unbiased view of the agreement. So, this figure isalso interesting when trying to automatize the agreements, that is, to carry outagreements between agents, virtual organizations or other pieces of software.Different approaches to automatic mediators can be seen in the literature suchas the PERSUADER system [12] or AutoMed [4].

5.1 Agreement Space as a CSP

As it was stated before, after the Agreement Space is defined, the different enti-ties involved in the agreement must fix (possibly through a negotiation process)some concrete values or intervals for the different dimensions comprising theAgreement Space. For such negotiation, each one of these entities will have someintervals of satisfying values for the different dimensions it is interested in. Thesesatisfying intervals may be expressed as constraints for the different dimensionsthat should be satisfied for the agreement. So, if this first phase of an agree-ment is seen from a mediator point-of-view, it can be seen as the solution of aConstraint-Satisfaction Problem (CSP), defined by the whole set of constraintsof the different entities. Being the Agreement Space n-dimensional, each one ofsuch constraints defines a (n − 1)-dimensional plane. As the Agreement Spaceis defined by the intersection of all such planes, it can be described how suchspace is by studying such planes. Therefore, this agreement mediator should beable to detect if an agreement is possible, checking if there exists an AgreementSpace as a result of all these constraint planes intersection.

The idea is to define the Agreement Space as the solution space of a CSPand to build an agreement by applying entities’ conditions as constraints inthe space (applying the HSA 6= [11] algorithm to build the hyperpolyhedronmodeling the CSP that will define the Agreement Space). A solution is reachedif all constraints are consistent. Furthermore, if the resulting space is convexthen it can be ensured that all interactions are inside the Agreement Space.

As it is an agreement, an Agreement Space (AS) is defined by a set of entitiesand their context (E, Cx), where the context is composed by variables, theirdomains an all the constraints. In this way, the AS is defined by extending acommon CSP definition.

5.2 Counselor Algorithm

The Counselor has the responsibility of helping the participants in the agreementto find the Agreement Space, considering it as a CSP. It is an iterative processthat adds new constraints one by one, guaranteeing that the new constraintis consistent with the previous ones. Constraints are inequalities of the form∑n

i=0 aixi ≤ b, where xi are the variables of the CSP. The set of all constraints

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defines an hyperplane and which union defines the limits for the valid values forall the variables involved in the agreement. The participants in the agreementwill take values inside the resulting hyperpolyhedron.

Using the mediated-agreement protocol defined in Section 4.2, the Coun-selor obtains the constraints from the participants and creates the AgreementSpace using a variant of HSA 6= algorithm. This algorithm is applied inside theAgreement Terms Definition subprotocol (see Figure 3b) each time a new con-straint Ci is received from a participant (by means of an Inform message). Sothe Agreement Space is incrementally constructed from scratch.

If Ci is inconsistent or redundant, the agent that proposed it is informed bya Reject message. The agent can redefine the constraint and submit a new oneif necessary.

Consistence and redundancy are checked using the HSA 6= algorithm. Ba-sically, when a new constraint Ci arrives the algorithm checks if each vertexof the current hyperpolyhedron satisfies Ci. If none vertex satisfy Ci then theconstraint is inconsistent and it is rejected. That is because the space defined bythe constraint does not contain any of the vertex of the hyperpolyhedron. Onthe other hand, if all vertex satisfy Ci then the constraint is redundant. That isbecause the complete hyperpolyhedron is included inside the space defined bythe constraint. In any other case, the constraint is consistent and it is added tothe hyperpolyhedron.

When all constraints are added to the Agreement Space and it is not emptythen there is an agreement among all the participants. If the Agreement Spaceis not empty, then it is ensured that it is convex, so any ’movement’ betweentwo points inside this space is always inside it. Therefore, it is guaranteed thatany negotiation process produced inside the agreement terms will never violatethe agreement.

The following section presents an example of reaching an agreement usinga Counselor Agent, interacting with the above presented interaction protocols,focusing in the Agreement-terms definition subprotocol.

6 Example

Let it be a set of agents interested in taking piano classes. The group is formedby one teacher and two students. They agree to negotiate over three dimensions:the number of classes (n), its duration (d) and its price (p). Initially, participantshave their own preferences (modeled as constraints).

– Teacher1. at least 10 classes: n ≥ 102. duration between 60 and 120 min: d ≥ 60 and d ≤ 1203. at least 20 euros/hour: p ≥ 20

– Student 11. no more than 20 classes: n ≤ 202. less than 30 euros/hour: p ≤ 30

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– Student 21. minimum 15 classes: n ≥ 152. duration between 45 and 90 min: d ≥ 45 and d ≤ 90

Assuming that agents have some common vocabulary (checked with the ADUprotocol) and they defined the above three dimensions as the agreement context,then the Counselor asks them for the terms of the agreement. There is no ruleabout the order in which agents propose the terms (constraints) nor the orderin which they arrive to the Counselor. Depending on it, one constraint can bedetected as redundant or just as a refinement of an existing one. But it does notaffect to the final Agreement Space.

Agent Mess Param Agent Mess Param Agent Mess Param

1 C inform n, d ,p 11 C call-for-terms 21 T inform d ≤ 120

2 C call-for-terms 12 T inform d ≥ 60 22 C accept d ≤ 120

3 T inform n ≥ 10 13 C accept d ≥ 60 23 C call-for-terms

4 C accept n ≥ 10 14 C call-for-terms 24 T inform p ≥ 20

5 C call-for-terms 15 S1 inform n ≤ 20 25 C accept p ≥ 20

6 S1 inform p ≤ 30 16 C accept n ≤ 20 26 C call-for-terms

7 C accept p ≤ 30 17 C call-for-terms 27 S2 inform d ≤ 90

8 C call-for-terms 18 S2 inform d ≥ 45 28 C accept d ≤ 90

9 S2 inform n ≥ 15 19 C reject d ≥ 45, red 29 C inform-done

10 C accept n ≥ 15 20 C call-for-terms

Table 1. Example of Agreement Terms Definition

This is a very simple case, in which restriction are defined over one dimension,but it is enough to follow and understand the process of construction of theAgreement Space. Let be C the counselor, T teacher, S1 student 1 and S2 student2. An example of the process can be followed in Table 1. In the sequence ofmessages on this table it can be seen how the Counselor communicates to therest of participants the context of the agreement (message 1). After that, theparticipants send their own constraints one by one (messages 3, 6, 9, 12, 15,18, 21, 24 and 27) and the Counselor accepts or rejects them. For example, inmessage 18, S2 proposes a constraint that is redundant, since one more restrictivehas been accepted in 12-13. But restrictions proposed in messages 15 and 27have been accepted because they refine (they are more restrictive) that the onesaccepted in 6-7 and 21-22 respectively.

7 Conclusions

Context Spaces seem to be a valid approach to model agreements and allowsto use well known techniques to solve the task of reaching an agreement andcontrolling its subsequent execution. The agreement is modeled as an object inthe Euclidean space.

The problem of defining the space associated to an agreement can be seen asa CSP, since the agreements conditions can be modeled as constraints and the

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agreement is expressed as a set of restrictions over a set of data. The conjunctionof all these restrictions will define an convex hyperpolyhedron delimiting theagreement space, that is, the space where all the computations fulfilling suchagreement will be carried out, the context of such computations.

Mediators can be used during the negotiation process to reach an agreementto check that any additional constraint keeps this space convex, along with con-trolling the feasibility of the agreement. For doing that, mediators can suggestmore relaxed or more strict constraints in order to maintain the consistence ofthe space and the possibility of an agreement.

This paper has presented not only the advantages of using such a Mediatoror Counselor in reaching an agreement, but also the way this Counselor worksand how to interact with him by means of some interaction protocols specificallydesigned for agreement reaching processes.

As future work, the dynamic of the agreement is going to be studied, becauseto guarantee an agreement, besides the existence of the agreement space, theconvergence of individual negotiation processes has to be guaranteed to avoidsystems that oscillate without ending in a concrete agreement for some initialsituations.

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